Interplanetary Mission Design

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Interplanetary Mission Design ASEN 5519 Spring
2008 Professor George H. Born ECNT
316 303-492-8638 georgeb_at_colorado.edu http//ccar.
colorado.edu/asen5519/imd/ Lecture 1
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Problem of Two Bodies
µ M1 M2
XYZ is nonrotating, with zero acceleration an
inertial reference frame
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Equations of Motion in the Orbit Plane
The u? component yields
which is simply h constant
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Solution of ur Equations of Motion

• The solution of the ur equation is (as function
  of ? instead of t)

where e and ? are constants of integration.
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Types of Orbital Motion
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The Orbit and Time

• If angle f is known, r can be determined from
  conic equation
• Time is preferred independent variable instead of
  f
• Introduce E, eccentric anomaly related to time
  t by Keplers Equation
• E e sin E M n (t tp)
• where M is mean anomaly

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Properties of Hyperbolic Orbits
asemimajor axis bsemiminor axis bangle of the
asymptote
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n-Body Problem

• Assume that we have a s/c denoted as m2 whose
  motion relative to m1 we wish to describe. Assume
  that the motion of m2 is perturbed by mj, other
  masses (planets) where

j3n
mj
r1j
From Newtons Law of Gravitation
m1
r1j
Gm1mj
r1j3
r2j
m1
r12
Z
r12
r1
Gm1m2
rj
r2j
r123
Gm2mj
m2
r2
r2j3
r12
O
- Gm1m2
Y
r123
X
inertial frame
m2
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n-Body Problem
From Newtons Second Law
so
Note that we must have and to solve
this equation. We may solve for by numerical
integration or by using a planetary ephemeris
file. Then
j3.n
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n-Body Problem
Notice that the n-body problem has 6 degrees of
freedom for each body or 6n total
degrees of freedom.
Hence, 6n constants of the motion are
required for a solution to the equations of
motion.
These constants are
n
Degrees of Freedom
Linear Momentum (CM)

Total Angular Momentum
Total Energy
vector
eliminates one component of the e-vector
Total Integrals of Motion (consts)
Jacobi constant included
velocity of c.g. is constant 6 consts are

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n-Body Problem

• Where
• Recall that , but
  since there are no external forces, is
  a constant.

The table on the previous slide is from Jeff
Parkers dissertation
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n-Body Problem

• The total energy of the system is given by
• The total work done by the external forces is
  equal to the change in total energy of the
  system. Because there are no external forces,
  total energy is conserved.

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Julian Day

• Calculation of the Julian Day No. at 0h UT Ref.
  (Curtis p214). Julian day is a continuous count
  from the year 4713 BC and begins at 12h UT, i.e.,
  Greenwich mean noon.
• Where Y, M, D lie in the following range

INT integer
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Julian Day

• At any other UT the Julian Day is given by
• where
• Example What is the Julian Day Number for Oct.
  4, 1957 UT 192624 (The launch date of
  Sputnick I)?

Jo Julian Day Number at 0 hr UT
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Julian Day
days
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Patch Conics

• The Patch Conic divides the planetary mission
  into three phases
• The Departure Phase Bodies are the Earth and
  S/C, the trajectory is a departure hyperbola with
  the Earth at the focus. Influence of the Sun and
  Target Planet are neglected.
• Cruise Phase Bodies are the Sun and S/C and the
  trajectory is a transfer ellipse with the Sun as
  the focus. Planets are neglected.
• Arrival Phase Bodies are target planet and S/C.
  The trajectory is an arrival hyperbola with the
  Planet at the focus. Sun and Earth are
  neglected.

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Patch Conics

• Phases begin and end at the sphere of influence
  (SOI). In reality, the SOI often is neglected
  since the effect is small. For the Earth, the
  true anomaly on the transfer ellipse should be
  reduced by 0.15 to account for the Earths SOI.

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Patch Conics

• Example Consider an Interplanetary mission from
  Earth to Venus.
• Assumptions
• 1) Planetary orbits are circular and coplaner.
• 2) Transfer ellipse is tangent to Earth Venus
  orbit i.e. and transfer is a simple
  Hohmann ellipse

Venus _at_ arrival
Earth _at_ launch
r?
?
?
?
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Patch Conics

• The transfer orbit elements are easily
  calculated

?
?
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Patch Conics
Transfer orbit elements contd Velocity at
aphelion, Velocity at perihelion,

, where

days
?
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Patch Conics
at departure on transfer ellipse
Hence, we must slow the heliocentric velocity of
the s/c
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Patch Conics
Normally we refer to the energy required from the
launch system as
at Venus, the target planet
?
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Patch Conics

• To capture the s/c at Venus we must supply a ?V
  to
• reduce velocity.
• Could choose an rp, compute the energy
• from , then compute periapsis velocity
  and
• circular velocity of capture orbit.
• This gives ?V needed for capture.